# Simple Regression

Chapter 14 Simple Linear Regression Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation for Estimation and Prediction Residual

Analysis: Validating Model Assumptions Outliers and Influential Observations 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 1 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Managerial decisions often are based on the relationship between two or more variables. Regression analysis can be used to develop an equation showing how the variables are related. The variable being predicted is called the dependent variable and is denoted by y. The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x.

2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 2 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Simple linear regression involves one independent variable and one dependent variable. The relationship between the two variables is approximated by a straight line. Regression analysis involving two or more independent variables is called multiple regression. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 3 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Model The equation that describes how y is related to x and an error term is called the regression model. The simple linear regression model is:

y = b0 + b1x +e where: b0 and b1 are called parameters of the model, e is a random variable called the error term. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 4 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Equation The simple linear regression equation is: E(y) = b0 + b1x Graph of the regression equation is a straight line. b0 is the y intercept of the regression line. b1 is the slope of the regression line. E(y) is the expected value of y for a given x value.

2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 5 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Equation Positive Linear Relationship E(y) Regression line Intercept b0 Slope b1 is positive x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide

6 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Equation Negative Linear Relationship E(y) Intercept b0Regression line Slope b1 is negative x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 7 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Equation

No Relationship E(y) Intercept Regression line b0 Slope b1 is 0 x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 8 or duplicated, or posted to a publicly accessible website, in whole or in part. Estimated Simple Linear Regression Equation The estimated simple linear regression

equation y b0 b1x The graph is called the estimated regression line. b0 is the y intercept of the line. b1 is the slope of the line. isythe estimated value of y for a given x value. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 9 or duplicated, or posted to a publicly accessible website, in whole or in part. Estimation Process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x

Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1 Sample x x1 . . xn Data: y y1 . . yn Estimated

Regression Equation y b0 b1x Sample Statistics b0, b1 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 10 or duplicated, or posted to a publicly accessible website, in whole or in part. Least Squares Method Least Squares Criterion min (yi y i )2 where: yi = observed value of the dependent variable for the ith observation yi =^estimated value of the dependent variable

for the ith observation 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 11 or duplicated, or posted to a publicly accessible website, in whole or in part. Least Squares Method Slope for the Estimated Regression Equation (x x)(y y) b (x x) i i 1

2 i where: xi = value of independent variable for ith observation yi = value of dependent variable for ith _ observation x = mean value for independent variable _ y = mean value for dependent variable 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 12 or duplicated, or posted to a publicly accessible website, in whole or in part. Least Squares Method y-Intercept for the Estimated Regression Equation

b0 y b1x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 13 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Example: Reed Auto Sales Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 14 or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Linear Regression Example: Reed Auto Sales Number of Number of TV Ads (x) Cars Sold (y) 1 14 3 24 2 18 1 17 3 27 Sx = 10 x 2

Sy = 100 y 20 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 15 or duplicated, or posted to a publicly accessible website, in whole or in part. Estimated Regression Equation Slope for the Estimated Regression Equation (xi x)(yi y) 20 b1 5 2 4 (xi x)

y-Intercept for the Estimated Regression Equation b y b x 20 5(2) 10 0 1 Estimated Regression Equation y 10 5x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 16 or duplicated, or posted to a publicly accessible website, in whole or in part. Coefficient of Determination

Relationship Among SST, SSR, SSE SST = SSE SSR + 2 2 2 ( y y ) ( y

y ) ( y y ) i i i i where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 17 or duplicated, or posted to a publicly accessible website, in whole or in part.

Coefficient of Determination The coefficient of determination is: r2 = SSR/SST where: SSR = sum of squares due to regression SST = total sum of squares 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 18 or duplicated, or posted to a publicly accessible website, in whole or in part. Coefficient of Determination r2 = SSR/SST = 100/114 = .8772 The regression relationship is very strong; 87.72% of the variability in the number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 19 or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Correlation Coefficient rxy (sign of b1 ) Coefficient of Determination rxy (sign of b1 ) r 2 where: b1 = the slope of the estimated regression equation y b0 b1 x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 20 or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Correlation Coefficient

rxy (sign of b1 ) r 2 y 10 5is x +. The sign of b1 in the equation rxy =+ .8772 rxy = +.9366 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 21 or duplicated, or posted to a publicly accessible website, in whole or in part. Assumptions About the Error Term e 1. The error e is a random variable with mean of zero 2. The variance of e , denoted by 2, is the same for all values of the independent variable. 3. The values of e are independent. 4. The error e is a normally distributed random

variable. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 22 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero. Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of 2, the variance of e in the regression model.

2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 23 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance An Estimate of 2 The mean square error (MSE) provides the estimate of 2, and the notation s2 is also used. s 2 = MSE = SSE/(n 2) where: SSE ( yi y i ) 2 ( yi b0 b1 xi ) 2 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 24

or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance An Estimate of To estimate we take the square root of 2. The resulting s is called the standard error of the estimate. SSE s MSE n 2 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 25 or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance: t Test Hypotheses H 0: b 1 0 H a: b 1 0 Test Statistic b1 t sb1 where sb1 s 2

S(xi x) 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 26 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: t Test Rejection Rule Reject H0 if p-value < or t < -t or t > t where: t is based on a t distribution with n - 2 degrees of freedom 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 27 or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance: t Test 1. Determine the hypotheses. H 0: b 1 0 H a: b 1 0 = .05 2. Specify the level of significance. b1 3. Select the test statistic.t sb1 4. State the rejection rule. Reject H0 if p-value < .05 or |t| > 3.182 (with 3 degrees of freedom) 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 28 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: t Test 5. Compute the value of the test statistic.

b1 5 t 4.63 sb1 1.08 6. Determine whether to reject H0. t = 4.541 provides an area of .01 in the upper tail. Hence, the p-value is less than .02. (Also, t = 4.63 > 3.182.) We can reject H0. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 29 or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence Interval for b1 We can use a 95% confidence interval for b1 to test the hypotheses just used in the t test. H0 is rejected if the hypothesized value of b1 is not included in the confidence interval for b1. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 30 or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence Interval for b1 The form of a confidence interval for b1 is: t / 2sb1 b1 t / 2sb1 is the margin of error b1 is the point t / 2 is the t value providing an area where estimat of /2 in the upper tail of a t distribution or

with n - 2 degrees of freedom 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 31 or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence Interval for b1 Rejection Rule Reject H0 if 0 is not included in the confidence interval for b1. 95% Confidence Interval for b1 b1 t / 2= sb15 +/- 3.182(1.08) = 5 +/- 3.44 or

1.56 to 8.44 Conclusion 0 is not included in the confidence interval. Reject H0 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 32 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: F Test Hypotheses H 0: b 1 0 H a: b 1 0

Test Statistic F = MSR/MSE 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 33 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: F Test Rejection Rule Reject H0 if p-value < or F > F where: F is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 34 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: F Test 1. Determine the hypotheses. H 0: b 1 0 H a: b 1 0 = .05 2. Specify the level of significance. 3. Select the test statistic.F = MSR/MSE 4. State the rejection rule. Reject H0 if p-value < .05 or F > 10.13 (with 1 d.f. in numerator and 3 d.f. in denominator) 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 35 or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Significance: F Test

5. Compute the value of the test statistic. F = MSR/MSE = 100/4.667 = 21.43 6. Determine whether to reject H0. F = 17.44 provides an area of .025 in the upper tail. Thus, the p-value corresponding to F = 21.43 is less than .025. Hence,evidence we rejectisHsufficient The statistical to 0. conclude that we have a significant relationship between the number of TV ads aired and the number of cars sold. Cengage Learning. All Rights Reserved. May not be scanned, copied 2014

Slide 36 or duplicated, or posted to a publicly accessible website, in whole or in part. Some Cautions about the Interpretation of Significance Tests Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a causeand-effect Just because we are able to reject H0: b1 = 0 and relationship is present between x and y. demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 37 or duplicated, or posted to a publicly accessible website, in whole or in part. Using the Estimated Regression Equation

for Estimation and Prediction A confidence interval is an interval estimate of the mean value of y for a given value of x. A prediction interval is used whenever we want to predict an individual value of y for a new observation corresponding to a given value of x. The margin of error is larger for a prediction interval. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 38 or duplicated, or posted to a publicly accessible website, in whole or in part. Using the Estimated Regression Equation for Estimation and Prediction Confidence Interval Estimate of E(y*) * t / 2sy* y

Prediction Interval Estimate of y* * y t / 2spred where: confidence coefficient is 1 - and t/2 is based on a t distribution with n - 2 degrees of freedom 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 39 or duplicated, or posted to a publicly accessible website, in whole or in part. Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: y =^ 10 + 5(3) = 25 cars 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 40 or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence Interval for E(y*) y*of Estimate of the Standard Deviation 1 (x* x)2 sy* s n (xi x)2 (3 2)2 1 sy* 2.16025 5 (1 2)2 (3 2)2 (2 2)2 (1 2)2 (3 2)2 sy* 2.16025 1 1 1.4491 5 4

2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 41 or duplicated, or posted to a publicly accessible website, in whole or in part. Confidence Interval for E(y*) The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: * t / 2sy* y 25 + 3.1824(1.4491) 25 + 4.61 20.39 to 29.61 cars 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 42

or duplicated, or posted to a publicly accessible website, in whole or in part. Prediction Interval for y* Estimate of the Standard Deviation of an Individual Value of y* 1 (x* x)2 spred s 1 n (xi x)2 1 1 spred 2.16025 1 5 4 spred 2.16025(1.20416) 2.6013 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 43 or duplicated, or posted to a publicly accessible website, in whole or in part. Prediction Interval for y*

The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: y* t / 2spred 25 + 3.1824(2.6013) 25 + 8.28 16.72 to 33.28 cars 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 44 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Analysis If the assumptions about the error term e appear questionable, the hypothesis tests about the significance of the regression relationship and the interval estimation results may not be valid. The residuals provide the best information about e . Residual for Observation i

yi yi Much of the residual analysis is based on an examination of graphical plots. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 45 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x If the assumption that the variance of e is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then The residual plot should give an overall impression of a horizontal band of points 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 46 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x y y Residual Good Pattern 0 x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 47 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x

Residual y y Nonconstant Variance 0 x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 48 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x y y Residual Model Form Not Adequate

0 x 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 49 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x Residuals Observation Predicted Cars Sold Residuals 1 15

-1 2 25 -1 3 20 -2 4 15 2 5

25 2 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 50 or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x TV Ads Residual Plot 3 Residuals 2 1 0 -1 -2

-3 0 1 2 TV Ads 3 4 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 51 or duplicated, or posted to a publicly accessible website, in whole or in part. Standardized Residuals Standardized Residual for Observation i

yi yi syi yi where: syi yi s 1 hi 1 (xi x)2 hi n (xi x)2 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 52 or duplicated, or posted to a publicly accessible website, in whole or in part. Standardized Residual Plot The standardized residual plot can provide

insight about the assumption that the error term e has a normal distribution. If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 53 or duplicated, or posted to a publicly accessible website, in whole or in part. Standardized Residual Plot Standardized Residuals Observation 1 Predicted y 15

Residual -1 Standardized Residual -0.5345 2 25 -1 -0.5345 3 20 -2

-1.0690 4 15 2 1.0690 5 25 2 1.0690 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 54 or duplicated, or posted to a publicly accessible website, in whole or in part.

Standardized Residual Plot Standardized Residual Plot A B Standard Residuals 28 29 30 31 32 33 34 35 36 37 C

1.5 D 1 RESIDUAL OUTPUT 0.5 Observation 0 -0.5 0 -1 -1.5 1 2 3 4 5

Predicted Y 15 10 25 20 15 25 Residuals Standard Residuals -1 -0.534522 20 30 -1 -0.534522 -2 -1.069045 2 1.069045 2 1.069045 Cars Sold 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied

Slide 55 or duplicated, or posted to a publicly accessible website, in whole or in part. Standardized Residual Plot All of the standardized residuals are between 1.5 and +1.5 indicating that there is no reason to question the assumption that e has a normal distribution. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 56 or duplicated, or posted to a publicly accessible website, in whole or in part. Outliers and Influential Observations Detecting Outliers An outlier is an observation that is unusual

in comparison with the other data. Minitab classifies an observation as an outlier if its standardized residual value is < orstandardized > +2. -2 This residual rule sometimes fails to identify an unusually large observation as being an outlier. This rules shortcoming can be circumvented by using studentized deleted residuals. The |i th studentized deleted residual| will be larger than the |i th standardized residual|. 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 57 or duplicated, or posted to a publicly accessible website, in whole or in part.

End of Chapter 14 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 58 or duplicated, or posted to a publicly accessible website, in whole or in part.

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